Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

We consider a class of linear matrix equations involving semi-infinite matrices which have a quasi-Toeplitz structure. These equations arise in different settings, mostly connected with PDEs or the study of Markov chains such as random walks on bidimensional lattices. We present the theory justifying the existence in an appropriate Banach algebra which is computationally treatable, and we propose several methods for their solutions. We show how to adapt the ADI iteration to this particular infinite dimensional setting, and how to construct rational Krylov methods. Convergence theory is discussed, and numerical experiments validate the proposed approaches

Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating $x$ when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) and $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, and $v$ obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of $x$. Pole selection strategies with explicit convergence bounds are given also in this case

Nonlocal PageRank

In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a \textit{long-range interaction} between them, does not exhibit concentration phenomena typical of spectral rankings which take into account just local interactions. We show that the predictive value of the rankings obtained using our proposals is considerably improved on different real world problems

Fast solvers for two-dimensional fractional diffusion equations using rank structured matrices

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either one-dimensional (1D) or 2D spatial domains. The use of an implicit Euler scheme in time and finite differences or finite elements in space leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats (H-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitz-like structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the use of rank-structured arithmetic is particularly effective and outperforms current state of the art techniques